Exploring Pi through Collisions

Aug 5, 2024

Mathematical Croquet: Exploring Collisions and Pi

Introduction

  • The lecture discusses a scenario involving two sliding blocks and a wall.
  • Assumptions:
    • No friction
    • Perfectly elastic collisions (no energy loss)
  • Aim: Count the number of collisions ("clacks") that occur in various scenarios.

Basic Scenario

  • Same Mass Blocks:
    • First block hits second block:
      • Transfers all momentum to the second block.
    • Second block bounces off the wall:
      • Transfers momentum back to the first block.
    • Total Clacks: 3

Different Mass Ratios

  • Case 1: First block 100 times the mass of the second:
    • Total collisions: 31
  • Case 2: First block 10,000 times the mass of the second:
    • Total collisions: 314
  • Case 3: First block 1,000,000 times the mass of the second:
    • Total collisions: 3,141

Pattern Identification

  • Observed pattern: When the mass of the first block is a power of 100 times the second, the number of collisions corresponds to the digits of pi.
  • Discovery Credit:
    • Originally discovered by mathematician Gregory Galperin in 1995, published in 2003.
    • Mentioned by viewer Henry Cavill.

Algorithm for Computing Pi

  • Steps to compute digits of pi using this phenomenon:
    1. Implement a physics engine.
    2. Choose the number of digits (d) of pi to compute.
    3. Set mass of first block to be 100^(d-1).
    4. Count collisions.
  • Example calculation for 20 digits:
    • Large block's mass = 100 billion billion billion billion times the mass of the smaller block (1 kg).
    • This results in counting 31 billion billion collisions.
    • Clack frequency: 100 billion billion billion billion clacks per second.

Implications and Observations

  • Theoretical implementation yields an impractical approach to computing pi.
  • Highlights the inefficiency of the method despite its elegance.
  • Raises the question: Why does pi appear in this context?
    • Pi is usually associated with circular geometry, hinting at a hidden circle in the problem.
  • The connection to conservation of energy will be explored in the next video.

Conclusion

  • Encouragement to explore the problem collaboratively.
  • Excitement for the next discussion about the underlying principles involving pi.
  • Thank you for watching!